Two ways to say a mapping from metric to metric is
continuous at point . The distance arguments are swapped compared
to metcnp18103 (and Munkres' metcn18105) for compatibility with df-lm16975.
Definition 1.3-3 of [Kreyszig] p. 20.
(Contributed by NM, 4-Jun-2007.)
(Revised by Mario Carneiro, 13-Nov-2013.)

Two ways to say a mapping from metric to metric is
continuous. Theorem 10.1 of [Munkres]
p. 127. The second biconditional
argument says that for every positive "epsilon" there is a
positive "delta" such that a distance less than delta in
maps to a distance less than epsilon in . (Contributed by NM,
15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

An absolute value
generates a metric defined by
, analogously to cnmet18297. (In fact, the
ring structure is not needed at all; the group properties abveq015607 and
abvtri15611, abvneg15615 are sufficient.) (Contributed by
Mario Carneiro,
9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Define a normed group, which is a group with a
right-translation-invariant metric. This is not a standard notion, but
is helpful as the most general context in which a metric-like norm makes
sense. (Contributed by Mario Carneiro, 2-Oct-2015.)

A normed ring is a ring with an induced topology and metric such that
the metric is translation-invariant and the norm (distance from 0) is an
absolute value on the ring. (Contributed by Mario Carneiro,
4-Oct-2015.)

Express the property of being a normed group purely in terms of
right-translation invariance of the metric instead of using the
definition of norm (which itself uses the metric). (Contributed by
Mario Carneiro, 29-Oct-2015.)